Functional nanoporous graphene superlattice

Two-dimensional (2D) superlattices, formed by stacking sublattices of 2D materials, have emerged as a powerful platform for tailoring and enhancing material properties beyond their intrinsic characteristics. However, conventional synthesis methods are limited to pristine 2D material sublattices, posing a significant practical challenge when it comes to stacking chemically modified sublattices. Here we report a chemical synthesis method that overcomes this challenge by creating a unique 2D graphene superlattice, stacking graphene sublattices with monodisperse, nanometer-sized, square-shaped pores and strategically doped elements at the pore edges. The resulting graphene superlattice exhibits remarkable correlations between quantum phases at both the electron and phonon levels, leading to diverse functionalities, such as electromagnetic shielding, energy harvesting, optoelectronics, and thermoelectrics. Overall, our findings not only provide chemical design principles for synthesizing and understanding functional 2D superlattices but also expand their enhanced functionality and extensive application potential compared to their pristine counterparts.


diffusion and bonding in graphene superlattice
In this study, we employed first-principles calculations based on DFT within the Vienna Ab initio Simulation Package (VASP) to investigate the diffusion energy barrier of tellurium (Te) atoms on the surface of a graphene superlattice 1,2 .To model the system, we constructed a supercell consisting of 263 carbon (C) atoms and one Te atom.Furthermore, we created two partially overlapped square-shaped pores, each with a diameter of 1.0 nm, by selectively removing carbon atoms from the layers of pristine bilayer graphene.The overlap ratio of the pores between the layers, defined as defined by the ratio of the pore area covered by a neighboring graphene layer to the total pore area, was set to 0.5.To ensure structural stability, we saturated the edges of the pores with hydrogen (H) atoms.The exchange and correlation interactions between the electrons were described using the Perdew-Burke-Ernzerhof (PBE) function within the generalized gradient approximation 3 .Additionally, we accounted for van der Waals interactions by incorporating the semi-empirical long-range dispersion correction proposed by Grimme, known as the DFT-D2 method.The plane-wave cutoff energy was set to 520 eV to achieve accurate and converged results.Total energies and force components were converged to 1×10 -5 eV and 1×10 -2 eV/Å, respectively.The reciprocal space was sampled using a Monkhorst-pack k-point mesh with a point density of 2×2×1.Fig. 1i in the main text represents two selected diffusion paths, along which we determined the diffusion energy barriers using the climbing image nudged elastic band (CI-NEB) method implemented in the VASP transition state tools 4,5 .Our results indicate that the diffusion of Te atoms towards the pore edge exhibits a lower energy barrier of 0.42 eV compared to diffusion towards the framework (0.76 eV).Furthermore, we analyzed the bonding between Te and C atoms at the pore edge and in the framework.
The Te-C bonding at the pore edge exhibited a significantly low negative binding energy of -7.41 eV, indicating a favorable interaction between Te and C atoms at the pore edges.In contrast, bonding of Te to C atoms in the framework yielded a positive binding energy of 0.72 eV.These findings suggest that achieving preferential element doping at the pore edges requires relatively low temperatures compared to doping within the framework 6 .In summary, our first-principles DFT calculations provide insights into the diffusion energy barriers and bonding behaviors of Te atoms in a graphene superlattice, which provide the potential for selective doping strategies and highlight the significance of pore edge effects in graphene superlattice structures.

Supplementary Note 2: Experimental evidence for electron domain wall in graphene superlattice
To gain insights into the mechanism underlying the presence of the electron domain wall, we performed an analysis of the crystal structure of the graphene superlattice using near-edge X-ray absorption fine structure (NEXAFS) characterization of the carbon atom K-edge.For comparison, we also measured pristine graphene and porous bilayer graphene with overlapped pores.The results, shown in Supplementary Fig. 10, reveal distinct features in the C K-edge spectrum of the graphene superlattice.
The presence of a G peak at 285.6 eV in the C K-edge spectrum of the graphene superlattice, lower than that of pristine graphene and porous bilayer graphene, indicates partial reconstruction of sp 2 C-C bonding in the graphene lattice due to the partial overlapping of carbon atoms 11,12 .This feature demonstrates a strong polarization dependence and suggests the formation of an electron domain wall.
Additionally, two peaks at 291.7 eV and 290.3 eV in the graphene superlattice spectrum are attributed to the 1s→σ* transition, indicating the presence of two types of disorders 13,14 .These disorders arise from in-plane strain and non-graphitized carbon-based bonds at the edges.Interestingly, the absence of a peak caused by lattice strain in both pristine graphene and porous bilayer graphene with overlapped pores suggests that lattice strain has been overlooked 15 .The combination of in-plane lattice strain and polarized C-C bonds in the graphene lattice collectively induces the formation of the electron domain wall.Furthermore, we observed that the G peak in the C K-edge spectrum of Te-doped graphene superlattice is further shifted to the left, and the intensity of the two disorder peaks is increased compared to the graphene superlattice.This indicates an enhanced in-plane strain and an increased intensity of polarized C-C bonding in the graphene lattice 16 .Overall, these findings provide experimental evidence for the presence of an electron domain wall in the graphene superlattice and shed light on its underlying structural characteristics.Supplementary Note 3: First-principles DFT simulation of electronic structure of graphene superlattice Here we utilized first-principles DFT to simulate the band structure of graphene.Our simulation process involved several steps, including crystal model design and optimization, building of a static self-consistent field, and calculation of the density of states 17,18 .All calculations were performed using VASP-6.1.0with the PBE function under the generalized gradient approximation.The DFT framework with D3 dispersion correction was combined with the projector augmented wave, and a plane wave cutoff energy of 500 eV was set 19 .The K-mesh for calculating the band structures of the samples was generated using vaspkit-1.2.3 with a line-mode consisting of 20 points.To prevent artificial interactions between periodic images, a 20 Å-thick vacuum layer was introduced perpendicular to the graphene sheet.The structures were relaxed until the residual forces on the atoms decreased to below 0.05 eV/Å.Phonon structures of graphene were calculated using the finite displacement method combined with phononpy-2.11 20.
In our study, we considered various factors, such as pore overlap ratio, doping amount and types, as well as other in-plane defect engineering methods, to comprehensively analyze their effects on the electronic band structure of graphene.

(i) Effect of nanopore overlap ratio
To investigate the impact of the pore overlap ratio on the electronic structure, we performed calculations on doped porous graphene systems with square-shaped nanopores with a diameter of 1.0 nm overlapping at different ratios: complete overlap (1), partial overlap (0.5), and no overlap (0).In these systems, the carbon atoms at the pore edge of the nanopore were saturated with an equal number of Te atoms.Pristine bilayer graphene was also included in the calculations for comparison.The electronic properties of graphene are primarily influenced by the behavior of the conduction and valence bands near the Fermi level.Therefore, these bands were analyzed due to their crucial role in determining the electronic characteristics of graphene.As shown in Supplementary Fig. 16, pristine graphene exhibits distinct conduction and valence bands with a cone-shaped structure that converges at the Dirac point.This cone-like dispersion signifies a strong linear relationship between electron energy and kinetic energy.However, there is no significant electronic density of states observed near the Fermi level.
When fully overlapped pores are introduced to bilayer graphene (the overlap ratio is 1), the Dirac point splits, resulting in the emergence of an open bandgap with a magnitude of 0.25 eV.Importantly, the band splitting does not significantly affect the electronic density of states near the Fermi level.With partial overlapping of the nanopores in bilayer graphene (the overlap ratio is 0.5), the band exhibits a remarkable flattening near the Fermi level, characterized by negligible band dispersion (less than 1 meV).This flat band structure arises from the weak dispersion relation of kinetic energy due to the periodic electronic reconstruction caused by atomic stress between the overlapped and exposed carbon regions 21,22 .This reconstruction leads to the formation of electron domain walls that confine Fermi electrons on either side, resulting in an equipotential Fermi surface and energy barriers with neighboring surfaces.The electron trapping effect leads to a closer proximity of valence electrons in the graphene superlattice to the Fermi level, resulting in the formation of multiple van Hoff singularities 22,23 , as demonstrated by the electron density distribution (Supplementary Fig. 12).When the nanopore is not overlapped (the overlap ratio is 0), the band shows a noticeable degree of dispersion (greater than 50 meV) compared to partially overlapping nanopores in bilayer graphene.However, it remains significantly weaker than the dispersion observed in pristine graphene and porous graphene with fully overlapped nanopores.It can be explained that in the case of completely non-overlapping nanopores, the graphene structure exhibits an increased area of exposed regions, in the form of monolayer graphene.
This enlarged area provides a platform for the release of interfacial strain that is generated between the overlapped regions of bilayer graphene and the exposed regions.Consequently, the electron domain wall at the interface is weakened, leading to a reduced ability to trap electrons.This weakening of the electron domain wall has a direct impact on the dispersion of the bands and limits their proximity to the Fermi level.As a result, lower electronic density of states near the Fermi level and non-negligible band dispersion are observed.

(ii) Effect of doping element concentration
In the case of doped graphene with partially overlapped nanopores (the overlap ratio is 0.5), we conducted further investigations to examine the influence of varying amounts of Te doping on the pore edges.Specifically, the carbon atoms at the edge of one pore were saturated with 2-10 Te atoms, while the remaining atoms were bonded with H.In the case of the graphene superlattice without Te doping, the pore edge was saturated with -OH, -COOH, and -OH bonds, which aligns with the graphene superlattice without edged doping.
As depicted in Supplementary Fig. 13, the band gradually turns to a flat shape as the Te atom content at the edges increases.When the number of Te atoms at the pore edge exceeds 8, the band dispersion becomes negligible (less than 1.0 meV).This observation indicates that high levels of Te doping at the pore edge effectively suppress the dispersion relationship between electron energy and kinetic energy, thereby contributing to the electron trapping effect 24 .Additionally, an increase in Te content leads to an upward shift of the valence band, resulting in the formation of one or multiple van Hoff singularities and a significant increase in the electronic density of states near the Fermi level.As the Te content further increases, the band dispersion becomes negligible, and the electron confinement becomes stronger, preventing the electrons from shifting further upward and confining them near the Fermi surface.Instead, the valence band located below the uppermost band continues to shift upward, leading to their overlapping and resulting in the merging of Fermi electrons with one strong van Hoff singularity near the Fermi level (exceeding 300 eV -1 ).

(iii) Effect of doping element types
In the case of bilayer graphene with partially overlapped nanopores (overlap ratio of 0.5), we conducted additional investigations to analyze the impact of different doping element types at the pore edges.Specifically, we saturated the carbon atoms at one pore edge with eight representative atoms of nitrogen (N), phosphorus (P), or sulfur (S), while the remaining atoms were bonded with H.The results depicted in Supplementary Fig. 14 demonstrate distinct effects of substituting the dopant element.When N is used as the dopant, a remarkable band dispersion exceeding 50 meV is observed.However, there is a low electronic density of states near the Fermi level, indicating a weak electron trapping effect and a lack of significant upward shift in the valence band.As we replace N with heavier elements such as P and S, the band dispersion gradually decreases with increasing atomic number.Additionally, the electronic density of states near the Fermi level experiences a notable increase.In the case of S substitution, two continuous van Hoff singularities emerge near the Fermi level with peak intensities surpassing 140 eV -1 .These intensities are approximately six times higher than that of N-doped graphene and three times higher than that of P-doped graphene, indicating a substantial upward shift in the valence band and pronounced merging of bands 25 .
The observed changes can be attributed to the energy difference between the bonding orbitals of the dopant atom (e.g., N for 2p, S and P for 3p) and the 2p orbital of the edge carbon.As the atomic number increases, this energy difference gradually enhances polarization.The enhanced polarization, facilitated by the orbital-spin coupling, leads to band splitting near the Fermi level and promotes an upward shift in the valence band 26,27 .Consequently, it increases the number of valence bands near the Fermi level, promotes the formation of multiple van Hoff singularities, and intensifies their strengths 28 .
The increased electronic density of states contributes to band flattening and enhances the electron trapping effect.However, when the energy difference exceeds a certain threshold, such as in the case of the 6p orbital of Te and the 2p orbital of C, the covalent bonds between the dopant atoms and the edge carbon can transform into coordination bonds 29 .This transformation weakens the strength of orbitalspin coupling due to weak overlapping of orbital wave functions 30 .Consequently, the intensities of the peaks and the electron trapping effect is somewhat diminished.

(iv) Effect of nanopore shape
We conducted further investigations to explore the influence of nanopore shape on bilayer graphene by considering partially overlapping circular nanopores.The overlap ratio was set at 0.5, and the edges of the pores were saturated with 8 Te atoms, following the same configuration as the Te-doped graphene superlattice mentioned earlier.As shown in Supplementary Fig. 15, the presence of circular nanopores leads to the splitting of the Dirac point, resulting in the emergence of a narrow band gap of approximately 0.1 eV.The underlying reason behind this behavior may lie in the structural characteristics of bilayer graphene with partially overlapping circular nanopores.It exhibits periodicity, overlapping, and exposed carbon atoms.However, the exposed region forms a curved nanoribbon with wider ends and a narrower waist, which introduces strain into the system.The generation of such strain disrupts the stress difference at the interface between the exposed and overlapped carbon atoms, making it challenging to effectively construct electron domain walls.Without electron domain walls, the confinement of electrons is hindered, and the generation of significant energy splitting is impeded.
Consequently, there is a lack of strong electron coupling, leading to a weak band dispersion 31 .In contrast to the flat band structure in graphene with square nanopores, the presence of circular nanopores in bilayer graphene with partial overlap does not result in a flat band structure.Instead, it exhibits a narrow band gap without significant electronic density of states near the Fermi level.

(v) Effect of in-plane defect engineering
Defect engineering is a widely used technique for manipulating the band structure of graphene, allowing precise control over its electronic properties.This approach involves introducing defects such as vacancies and specific element doping.Additionally, the electronic structure of graphene can be customized by creating graphene nanoribbons with controlled dimensions.Here we examined three types of defected graphene: Te-doped pristine graphene, graphene with vacancy defects (e.g., a removed carbon atom), and an armchair graphene nanoribbon measuring 2.0 nm in length and 0.5 nm in width.
The edges of the graphene nanoribbon were saturated with H atoms.As depicted in Supplementary Fig. 16, all three types of graphene demonstrate distinct dispersion behavior in their band structures, accompanied by varying degrees of Dirac point splitting, ultimately leading to the formation of bandgaps.Furthermore, there is no significant electronic density of states observed near the Fermi level.
These observations can be attributed to the absence of periodic electron domain walls within the graphene layers, resulting in a negligible electron coupling effect 32,33 .

Supplementary Note 4: Molecular dynamics simulation of the phonon coupling of graphene superlattice
To evaluate the degree of phonon coupling in the graphene superlattice and assess the compatibility of the newly generated low-frequency phonons, molecular dynamics simulations were conducted using the open-source code large-scale atomic/molecular massively parallel simulator (LAMMPS) [34][35][36] .The computational model was designed to align with experimental data, and square pores with an overlap ratio of 0.5 was chosen for the graphene superlattice system.The square nanopores in the simulation had a diameter of 7.0 nm, and the width between nanopores on each atomic layer was set to 1.0 nm.
The simulation process involved initial energy minimization, followed by relaxation under the constant pressure and temperature (NPT) ensemble at 300 K for 50 ps, with a time step of 1 fs.Subsequently, the simulation continued at 300 K for an additional 100 ps under the constant volume and temperature (NVT) ensemble.During the last 50 ps of the simulation, the velocities of the carbon atoms of interest were sampled every 5 fs to calculate the vibrational density of states [37][38][39] .
To accurately model the interactions between the exposed and overlapped carbon atoms in the graphene superlattice, the adaptive intermolecular reactive empirical bond order (AIREBO) potential was employed 40 .The classification of carbon atoms within the system was based on their proximity to the pore, where carbon atoms within a distance smaller than 10 Å were identified as pore edge carbon atoms, while carbon atoms beyond this distance were considered as carbon atoms within the framework.
The vibrational density of states, denoted as P(ω), was calculated as 41 : where ω and vj(t) represent the angular frequency and velocity of carbon atom j at time t, respectively.
The ensemble average in Equation S1 was approximated by a time average computed over a period of 50 ps once the system reached equilibrium.The coupling degree between any two calculated vibrational density of states (P(ω)) can be determined by 42 : The results depicted in Supplementary Fig. 17 demonstrate that the phonon coupling between the carbon atoms at the pore edge and in the framework is relatively weak, with a value as low as 0.71.Similarly, the coupling degree between the overlapped carbon atom and the exposed carbon atom is even lower, reaching a value as low as 0.51.These findings indicate poor compatibility among the newly generated phonons, resulting in increased scattering between them.Consequently, the phonon thermal conductivity is significantly reduced, leading to a decrease in the overall thermal conductivity.
Supplementary Note 7: The correlation between electron and phonon structures in extended applications (i) Connection of electron structure with EM modulation and luminescence performance: Observations of the graphene superlattice's band indicate multiple van Hoff singularities near the Fermi level (Fig. 2a).These singularities stem from electron domains that split the Fermi surface, creating new Fermi surfaces with differing energy levels.This property enables the graphene superlattice to harness external electric fields to excite Fermi electrons, consequently enhancing conductivity and linearly influencing the permittivity.The linear increase in permittivity indicates the film's suitability for EM wave transmission, absorption, and shielding as it progressively rises, suggesting EM switchability (Fig. 4a).The increased energy level splitting and potential differences across the electron domain wall lead to the separation of electrons and holes, producing a notable photoluminescence effect in the visible light wavelength range (400-700 nm), distinct to the graphene superlattice and not observed in pristine graphene (Fig. 4b).
(ii) Connection of phonon structure with reduced thermal conductivity: Upon examining the impact of the superlattice structure on the graphene phonon structure, it becomes apparent that pristine graphene predominantly features atom-propagated phonons at a frequency of 50 THz (Fig. 2c).However, in bilayer graphene superlattices, the periodic overlap of carbon atoms hinders harmonic vibrations, creating distinct phonon modes in the overlapped and exposed regions.This unique transmission of phonon clusters results in significant coherent interference and elastic scattering, weakening the energy of the phonon clusters.This generates a series of continuous standing waves in the low-frequency range of 10-30 THz, substantially reducing the phonon mean free path.Consequently, this reduction leads to notable localization and a decline in thermal infrared emission (Fig. 4c and Supplementary Fig. 28).
(iii) Connection of electron-phonon Interaction with thermoelectric behavior: The localization of low-frequency phonons and the electron trapping effect causes a strong electron-phonon coupling.As a result, electrons absorb phonons and scatter to excite new phonons, initiating a pronounced phonon drag effect.This effect significantly impacts the trajectories and transport properties of electrons within the graphene superlattice.This substantial phonon drag positively influences the Seebeck coefficient (Fig. 4d).The amplified Seebeck coefficient, combined with reduced thermal conductivity, results in a nearly two orders of magnitude higher ZT value compared to pristine graphene, showcasing its exceptional promise for thermoelectric applications (Fig. 4f).
direction, a larger energy barrier, with insulator-like characteristics, is present, leading to a weaker Seebeck effect.
Our investigation into the remarkable in-plane thermoelectric enhancement and relatively limited effects in the out-of-plane direction focused on pivotal structural modifications.These include the strategic creation of nanopores and selective edge doping within the plane, differing from conventional methods that alter interlayer interactions.Our synthetic method significantly modified in-plane electron and phonon structures, creating phenomena such as electron traps, coherent interference, elastic phonon scattering, and robust electron-phonon coupling.These collective alterations notably elevated the inplane ZT value.While exploring interlayer dynamics, adjustments such as nanopore introduction or element doping had limited impact on fundamental interlayer forces, resulting in only a slight improvement in out-of-plane thermoelectric properties when compared to pristine graphene.